The artifacts were samples from a 5,500-piece collection that dealers had offered to sell Mr. Green for $2 million. He was in Dubai with a paid adviser in the summer of 2010, seeking relics to install in what would become the $500 million Museum of the Bible in Washington, D.C.

After months of back and forth, Mr. Green’s family business paid $1.6 million for the lot. Mr. Green thought he was getting a bargain.

Instead, the deal cost Hobby Lobby another $3 million in July to settle a government lawsuit alleging the goods were smuggled into the U.S. Most of the relics were seized by federal authorities, and Iraq has since petitioned the Justice Department to return items that Iraqi officials believe were stolen from their country.

Archaeologists and biblical scholars now say there are questions about the origins and authenticity of the rest of the Green collection. That has scared away some researchers from the collection and the Bible museum over concerns about working with possibly looted antiquities. It has also cast a shadow over the Nov. 17 opening of the museum the Green family built, in part to house its collection.

“The Greens are good people, and they wanted to find important artifacts to study, but the situation may end up hurting some of the scholars and students they wanted to help,” said Josephine Dru. She was curator of the collection’s papyri for more than three years until she left last fall, in part, over the museum’s handling of manuscripts with no clear ownership records.

Mr. Green and his family have amassed a $205 million collection of roughly 40,000 artifacts over the past eight years, spanning Egyptian mummy masks from the time of Moses to sacred scrolls inscribed by rabbis and monks to a copy of the Bible once owned by Elvis Presley.

The collection was planned for inclusion in the 430,000 square-foot Museum of the Bible. That’s as big as the Smithsonian’s new National Museum of African American History and Culture.

Yet for the past three years, staff members at the museum have been sifting and rejecting potential donations from the Green collection that carried lingering questions of provenance, its director David Trobisch said. Most of the Greens’ antiquities remain with Hobby Lobby, the family owned arts-and-crafts chain.

Mr. Green said his 2010 purchase in Dubai was the misstep of an amateur, an unfortunate mix of enthusiasm and inexperience. Boxes of the items shipped to the Hobby Lobby headquarters in Oklahoma were mislabeled as “ceramic tiles,” and several boxes slipped undetected past U.S. customs inspectors before the scheme was uncovered, according to authorities.

“We made mistakes, but we’ve learned from them,“ said Mr. Green, 54 years old. Mr. Green said he had no idea the deal would imperil the reputation of the museum he long championed.

Among academicians, the matter remains unresolved. Bible scholars Candida Moss at the University of Birmingham and Joel Baden of Yale have just published a book, “Bible Nation: The United States of Hobby Lobby,” that said Mr. Green acquired objects with a “naiveté that begins to look willful.”

Roberta Mazza, a papyrologist at the University of Manchester, has criticized Mr. Green’s collecting methods in lectures at scholarly conferences on art crime and biblical literature. She said she noticed an ancient Coptic fragment from the New Testament Book of Galatians, displayed at an exhibit of the Green collection three years ago, which she believed she had earlier seen for sale on eBay. It offered no ownership history, she said.

“I don’t care if a billionaire wants to open a museum so long as it’s ethical,” Ms. Mazza said.

The Bible museum’s director, Mr. Trobisch, said the Green collection’s records indicate the family didn’t buy the Coptic fragment on eBay. Still, he said, the museum has no plans to exhibit it “until we can research it thoroughly.”

Mr. Trobisch recently advised Mr. Green to repatriate the fragment to Egypt, which Mr. Green said he would consider.

When the museum opens, its permanent collection will consist of just 2,840 vetted objects, Mr. Trobisch said, a fraction of what the Greens own. The museum has had to distance itself from controversy over the collection, he said: “We feel like we’re paying for the sins of the father.”

Jana Mathews, a medievalist at Rollins College, said scholars insist on knowing the provenance of objects because demands for proper paperwork discourages looting.

The museum has shifted its focus from displays of ancient art and artifacts to more interactive, theme-park experiences. Rooms will give a firsthand view of life in biblical times, including a life-size village in first-century Galilee. Elsewhere, walk-through exhibits will re-enact famous Bible stories—like Moses parting the Red Sea. An aerial-simulation ride will zoom passengers around Washington, D.C., to view scriptures inscribed on buildings.

Museum of the Bible Vice Chairman Robert Cooley said the board learned about the government’s six-year smuggling investigation only when Hobby Lobby was close to signing the settlement. He said the museum board felt “shortchanged on the facts” and has since hired its own lawyer to help the board toughen its provenance protocol and keep close watch on acquisitions.

Mr. Green said he didn’t tell the board sooner because he considered it a Hobby Lobby matter.

Last month, the board also hired its own cultural-heritage lawyer, Thomas Kline, to vet the pieces remaining in the museum’s collection.

“Before this, we weren’t collectors or museum-goers,” Mr. Green said. “We didn’t know we needed to ask for all this paperwork.”

The grandson of a Pentecostal preacher, Mr. Green grew up in the small town of Bethany, Okla. His father, David, transformed a picture-frame business into a chain of more than 750 retail stores. The elder Mr. Green remains chief executive of the family business, which last year had $4.4 billion in sales.

Steve Green married young and joined the family business. He stepped onto the public stage after the company won a Supreme Court case three years ago to exempt it from a federal mandate to pay for certain contraceptives for its workers. The family had argued the requirement conflicted with its Christian faith.

For Mr. Green, who remains influential in evangelical circles, the dream of building a Bible museum helped inspire him to collect antiquities. He said he knew from the start that he would need expert help.

In 2009, Mr. Green started working with Scott Carroll, who had been the adviser for Robert Van Kampen, a Chicago investor who had the largest private Bible collection in the U.S. at the time. That collection became part of a religious theme park in Orlando, Fla., called the Holy Land Experience.

Mr. Carroll, who has a doctorate in history from Ohio’s Miami University, was controversial in the field of antiquities for using a soapy solution to break apart the papyrus layers that form mummy masks. In a 2012 video, filmed by a research arm of the Green collection, Mr. Carroll washes a kohl-eyed, papery mask in a sink, then peels it apart looking for scraps of legible writing. “Archaeology is a destructive science,” he tells onlookers in a later video.

Holger Strutwolf, director of the Institute for New Testament Textual Research at the Bible Museum in Münster, Germany, said, “The way Scott Carroll treats mummy cartonnage comes close to willful damage.” Mr. Carroll said other scholars have dismantled mummy masks.

With Mr. Carroll on salary, the Green family went on a buying spree, adding roughly 18 objects a day to their collection, a breathtaking pace for antiquities.

Mr. Carroll said his job was to flag potential purchases. The Green family decided at closed-door meetings whether to buy. Often, Mr. Carroll said, he “had no idea an acquisition had been made until the items showed up” at company headquarters. Mr. Green didn’t dispute the description.

Cary Summers, the Bible museum president, said Mr. Carroll was encouraged to seek out large private collections, which included a 10,000-piece lot of biblical Americana in Maryland. Museums typically cherry-pick from private holdings rather than buy in bulk.

Mr. Green accompanied Mr. Carroll on a few buying trips to Jerusalem and Europe between 2009 and 2011. The sellers included some of Jerusalem’s biggest antiquities dealers, Mr. Green said, mostly families who had run shops for several generations.

After the Dubai trip in 2010, Mr. Carroll said he twice told Mr. Green to end the purchase negotiations because of “issues of provenance” with the cuneiform tablets. He said Mr. Green told him, “My family is not averse to risk.” Mr. Green didn’t dispute Mr. Carroll’s account.

Israeli authorities later said the dealers in Dubai churned out fake invoices and concocted histories for the Iraqi antiquities they sold to Mr. Green. Israel has since charged five dealers in Jerusalem with tax fraud for their part in orchestrating the deal. Emails to the men seeking comment weren’t returned.

In summer 2010, a Hobby Lobby lawyer invited a cultural-heritage law expert from the DePaul University College of Law, Patty Gerstenblith, to give a presentation to Mr. Green, Mr. Carroll and others on the best practices for collecting antiquities. Mr. Summers, the museum’s president, said later it was intended to help Messrs. Green and Carroll steer clear of legal trouble.

Ms. Gerstenblith said a staff member told her Mr. Green was considering buying pieces from Iraq. She emailed a report to Hobby Lobby that summarized her presentation.

“An estimated 200,000 to 500,000 objects have been looted from archaeological sites in Iraq since the early 1990s; particularly popular on the market and likely to have been looted are cylinder seals, cuneiform tablets,” said the report, which was in court documents. “Any object brought into the U.S. with Iraq declared as country of origin has a high chance of being detained by U.S. Customs.”

On Dec. 8, 2010, Mr. Green, through Hobby Lobby, agreed to buy the artifacts he had seen in Dubai. One of the dealers suggested shipping the pieces via FedEx. Mr. Green’s executive assistant agreed and wrote on Dec. 23 “as long as you keep the value of each package under $2,000, they will not have to forward it to our broker to clear it through Customs,” court documents said.

Mr. Carroll said he didn’t advise Mr. Green’s assistant on the matter and that he was shocked the Greens proceeded with the deal after Ms. Gerstenblith’s presentation.

Three days later, the dealer falsified the labels and shipped eight packages to Hobby Lobby, each valued from $250 to $300. The first three sailed through customs in Memphis, Tenn.; the next five were stopped and triggered the government investigation.

By 2012, Mr. Summers said, Mr. Green decided to fire Mr. Carroll, who left behind scant notes on the provenance of thousands of purchases, according to two curators assigned to review them.

Mr. Carroll said the Greens could have required the dealers to submit relics’ ownership history. “Admittedly,” he said, “it was not collected by me.”